zero-coupon bond. Addison Company will issue a zero-coupon bond this coming month. The bond's projected yield is \( 6 \% \). If the par value is \( \$ 1,000 \), what is the bond's price using a semiannual convention if a. the maturity is 20 years? b. the maturity is 35 years? c. the maturity is 70 years? d. the maturity is 100 years? a. What is the price of the bond using a semiannual convention if the maturity is 20 years? \( \$ \square \) (Round to the nearest cent.)

To calculate the price of a zero-coupon bond using a semiannual convention, we can use the formula: \[ P = \frac{F}{(1 + r/n)^{nt}} \] Where: - \( P \) = price of the bond - \( F \) = par value of the bond - \( r \) = annual yield (as a decimal) - \( n \) = number of compounding periods per year - \( t \) = number of years to maturity Given: - \( F = 1000 \) - \( r = 0.06 \) - \( n = 2 \) (since it's semiannual) - \( t \) will vary based on the maturity. ### a. Maturity of 20 years 1. Substitute the values into the formula: - \( t = 20 \) \[ P = \frac{1000}{(1 + 0.06/2)^{2 \times 20}} \] 2. Calculate \( P \): Let's perform the calculation. First, calculate \( r/n \): \[ \frac{0.06}{2} = 0.03 \] Now calculate \( nt \): \[ 2 \times 20 = 40 \] Now substitute these values into the formula: \[ P = \frac{1000}{(1 + 0.03)^{40}} \] Now, we will calculate \( (1 + 0.03)^{40} \) and then find \( P \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(\left(1+0.03\right)^{40}\) - step1: Add the numbers: \(1.03^{40}\) - step2: Convert the expressions: \(\left(\frac{103}{100}\right)^{40}\) - step3: Use the properties of exponents: \(\frac{103^{40}}{100^{40}}\) Calculate or simplify the expression \( 1000/(1 + 0.03)^{40} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{40}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{40}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{40}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{40}}{100^{40}}}\) - step4: Rewrite the expression: \(\frac{10^{83}}{103^{40}}\) The price of the zero-coupon bond with a maturity of 20 years is approximately: \[ P \approx 306.56 \] Thus, the price of the bond is: \[ \$ 306.56 \] ### Now, let's calculate the price for the other maturities: ### b. Maturity of 35 years 1. Substitute \( t = 35 \) into the formula: \[ P = \frac{1000}{(1 + 0.03)^{2 \times 35}} \] 2. Calculate \( nt \): \[ 2 \times 35 = 70 \] Now substitute into the formula: \[ P = \frac{1000}{(1 + 0.03)^{70}} \] ### c. Maturity of 70 years 1. Substitute \( t = 70 \) into the formula: \[ P = \frac{1000}{(1 + 0.03)^{2 \times 70}} \] 2. Calculate \( nt \): \[ 2 \times 70 = 140 \] Now substitute into the formula: \[ P = \frac{1000}{(1 + 0.03)^{140}} \] ### d. Maturity of 100 years 1. Substitute \( t = 100 \) into the formula: \[ P = \frac{1000}{(1 + 0.03)^{2 \times 100}} \] 2. Calculate \( nt \): \[ 2 \times 100 = 200 \] Now substitute into the formula: \[ P = \frac{1000}{(1 + 0.03)^{200}} \] Now, let's perform the calculations for maturities of 35, 70, and 100 years. Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{200}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{200}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{200}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{200}}{100^{200}}}\) - step4: Rewrite the expression: \(\frac{10^{403}}{103^{200}}\) Calculate or simplify the expression \( 1000/(1 + 0.03)^{140} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{140}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{140}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{140}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{140}}{100^{140}}}\) - step4: Rewrite the expression: \(\frac{10^{283}}{103^{140}}\) Calculate or simplify the expression \( 1000/(1 + 0.03)^{70} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1000}{\left(1+0.03\right)^{70}}\) - step1: Add the numbers: \(\frac{1000}{1.03^{70}}\) - step2: Convert the expressions: \(\frac{1000}{\left(\frac{103}{100}\right)^{70}}\) - step3: Simplify the expression: \(\frac{1000}{\frac{103^{70}}{100^{70}}}\) - step4: Rewrite the expression: \(\frac{10^{143}}{103^{70}}\) Here are the calculated prices for the zero-coupon bond at different maturities: ### b. Maturity of 35 years The price of the bond is approximately: \[ P \approx 126.30 \] Thus, the price of the bond is: \[ \$ 126.30 \] ### c. Maturity of 70 years The price of the bond is approximately: \[ P \approx 15.95 \] Thus, the price of the bond is: \[ \$ 15.95 \] ### d. Maturity of 100 years The price of the bond is approximately: \[ P \approx 2.71 \] Thus, the price of the bond is: \[ \$ 2.71 \] ### Summary of Prices: - Maturity of 20 years: \$306.56 - Maturity of 35 years: \$126.30 - Maturity of 70 years: \$15.95 - Maturity of 100 years: \$2.71